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Central difference approximationsfor otherderivatives can beobtained from
Eqs. (a)-(h) in a similar manner. For example, eliminating
f
(
x
)fromEqs. (f ) and (h)
and solving for
f
(
x
) yield
f
(
x
+
2
h
)
−
2
f
(
x
+
h
)
+
2
f
(
x
−
h
)
−
f
(
x
−
2
h
)
f
(
x
)
(
h
2
)
=
+
O
(5.3)
2
h
3
The approximation
+
−
+
+
−
−
+
−
f
(
x
2
h
)
4
f
(
x
h
)
6
f
(
x
)
4
f
(
x
h
)
f
(
x
2
h
)
f
(4)
(
x
)
(
h
2
)
=
+
O
(5.4)
h
4
is available fromEq. (e) and (g) after eliminating
f
(
x
). Table 5.1 summarizes the
results.
−
−
+
+
f
(
x
2
h
)
f
(
x
h
)
f
(
x
)
f
(
x
h
)
f
(
x
2
h
)
2
hf
(
x
)
−
1
0
1
h
2
f
(
x
)
1
−
2
1
2
h
3
f
(
x
)
−
1
2
0
−
2
1
h
4
f
(4)
(
x
)
1
−
4
6
−
4
1
Table 5.1.
Coefficients of central finite difference approximations
of
O
(
h
2
)
First Noncentral Finite Difference Approximations
Central finite difference approximations are not always usable.For example, consider
the situationwhere the functionis givenat the
n
discrete points
x
1
,
x
2
,...,
x
n
.Since
central differences use values of the function on each sideof
x
, we wouldbe unable to
compute the derivatives at
x
1
and
x
n
.Clearly, there is aneed for finite difference
expressionsthat requireevaluations of the function only on onesideof
x
. These
expressions arecalled
forward
and
backward
finite difference approximations.
Noncentral finite differences can also beobtained fromEqs. (a)-(h).Solving
Eq. (a)for
f
(
x
) we get
h
2
6
h
3
4!
f
(
x
+
h
)
−
f
(
x
)
h
2
f
(
x
)
f
(
x
)
f
(
x
)
f
(4)
(
x
)
=
−
−
−
−···
h
Keeping only the first term on the right-hand sideleads to the
first forward difference
approximation
f
(
x
+
h
)
−
f
(
x
)
f
(
x
)
=
+
O
(
h
)
(5.5)
h
Similarly, Eq. (b) yields the
first backward difference approximation
f
(
x
)
−
f
(
x
−
h
)
f
(
x
)
=
+
O
(
h
)
(5.6)
h
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